Calculus I: The Product Rule (Level 2 of 3)

This video tutorial covers advanced examples of using the product rule in calculus. It begins with a basic introduction to the product rule, followed by examples involving differentiable functions, power functions, rational functions, and exponential functions. The video also demonstrates how to evaluate derivatives at specific points using given function values. The tutorial concludes with a summary and a preview of the next topic, which involves using graphs to find derivatives.

What is the product rule used for in calculus?

• Finding the integral of a function

• Calculating limits

• Finding the derivative of a product of two functions

• Solving differential equations

In the first example, what are the two functions whose product's derivative is being found?

• f(x) and g(x)

• a(x) and b(x)

• x and y

• p(x) and q(x)

When applying the product rule to R(x) = x^2 q(x) + p(x), what is the derivative of the first term?

• x^2 q'(x) + q(x) 2x

• 2x q(x) + x^2 q'(x)

• x^2 + q(x)

• q(x) + 2x

In the example with R(x) = q(x)/x + e^x p(x) + 5x, what is the derivative of the term q(x)/x?

• q'(x) - x

• q(x) + x

• q(x)/x^2 - q'(x)/x

• -q(x)/x^2 + q'(x)/x

What is the derivative of the term e^x p(x) in the example with R(x) = q(x)/x + e^x p(x) + 5x?

• p(x) e^x

• e^x p'(x) + e^x p(x)

• e^x + p(x)

• p'(x) e^x

In the problem where f(x) = g(x)h(x), what is the value of f'(2) given g(2) = 3, g'(2) = -2, h(2) = -1, and h'(2) = 4?

• 10

• 14

• 12

• 16

What is the expression for f'(x) when f(x) = g(x)h(x)?

• g(x) + h(x)

• g'(x)h(x) + h'(x)g(x)

• g'(x) + h'(x)

• g(x)h'(x) + h(x)g'(x)

In the example where f(x) = e^x h(x), what is the value of f'(0) given h(0) = 5 and h'(0) = 2?

• 7

• 9

• 11

• 5

What is the derivative of f(x) = e^x h(x) using the product rule?

• e^x h'(x) + h(x) e^x

• h'(x) e^x

• h(x) e^x

• e^x + h(x)

What is the key concept emphasized in the final example of the video?

• Finding limits

• Solving integrals

• Applying the product rule with given values

• Using graphs to find derivatives